We study numerically a strongly nonlinear long wave model for surface gravity waves propagating in both one and two horizontal dimensions. This model often referred to as the Su-Gardner or Green-Naghdi equations can be derived from the Euler equations under the assumption that the ratio between the characteristic wavelength and water depth is small, but no assumption on the wave amplitude is required. We first generalize the model to describe large amplitude one-dimensional solitary waves in the presence of background shear of constant vorticity. After computing the solitary wave solution of the strongly nonlinear model, the interaction between two solitary waves propagating in the same and opposite directions is investigated numerically and the numerical solutions are compared with weakly nonlinear asymptotic solutions. In particular, the effects of strong nonlinearity as well as background shear are examined. We also derive a model for strongly nonlinear long waves in uniform shear flow interacting with non-uniform bottom topography, and the generation of upstream-propagating solitary waves is investigated numerically. We then examine the stability characteristics of large amplitude solitary waves subject to transverse perturbations with assuming that the characteristic wavelength in the transverse direction is much greater than that in the wave propagation direction. Using an asymptotic approach, a sufficient condition for instability is obtained. To test this result, we solve numerically the strongly nonlinear, weakly two-dimensional model using a finite difference method. This numerical model is also used to study the interaction between two solitary waves propagating obliquely with a small angle. In particular, the Mach reflection due to the strong oblique interaction is investigated numerically in detail and our numerical solution is compared with the analytical solution of the weakly nonlinear KP equation as well as available experimental data.